I’m a big walker, and lately one of my favorite walks has been from

Lately, I’ve noticed a strange

But there’s a problem—while I’m going on these walks, I’m daydreaming, thinking about random things, calling friends and family, or so on. And even if I wasn’t doing any of those things, I have a tendency to forget a

Let’s

Modelling the Process

Let’s denote by $S_t$ the running average of my walk, after taking $t$ samples. Of course, I don’t know $t$—I just know that, whatever $t$ I’m on, I

\[\mathbb{E}[S_{t+1} \mid S_t] = p \cdot (\mathbb{E}[S_t] + 1) + q \cdot (\mathbb{E}[S_t] - 1) = \mathbb{E}[S_t] + (p - q).\]

Under the null hypothesis, $p=q=1/2$, and so $S_t$ is a martingale. Intuitively, we think that this

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Theorem (Ville’s Inequality)

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This is a general fact (which extends way beyond real vector spaces), but it has a really nice implication for us. It turns out, we can actually define the gradient relative to this representation:

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In any case, thanks for reading.

—Thomas